The lifespans of porcupines in a particular zoo are normally distributed. The average porcupine lives $18.2$ years; the standard deviation is $2.4$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a porcupine living less than $11$ years.
$18.2$ $15.8$ $20.6$ $13.4$ $23$ $11$ $25.4$ $99.7\%$ $0.15\%$ $0.15\%$ We know the lifespans are normally distributed with an average lifespan of $18.2$ years. We know the standard deviation is $2.4$ years, so one standard deviation below the mean is $15.8$ years and one standard deviation above the mean is $20.6$ years. Two standard deviations below the mean is $13.4$ years and two standard deviations above the mean is $23$ years. Three standard deviations below the mean is $11$ years and three standard deviations above the mean is $25.4$ years. We are interested in the probability of a porcupine living less than $11$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $99.7\%$ of the porcupines will have lifespans within 3 standard deviations of the average lifespan. The remaining $0.3\%$ of the porcupines will have lifespans that fall outside the shaded area. Because the normal distribution is symmetrical, half $({0.15\%})$ will live less than $11$ years and the other half $({0.15\%})$ will live longer than $25.4$ years. The probability of a particular porcupine living less than $11$ years is ${0.15\%}$.